Galois Representations with Quaternion Multiplications Associated to Noncongruence Modular Forms

Details Galois Representations with Quaternion Multiplications Associated to Noncongruence Modular Forms Galois Representations with Quaternion Multiplications Associated to Noncongruence Modular Forms

2010-05-22

arxiv.org/abs/1005.4105v4

Mathematics

Number Theory

Scientific paper

In this paper we study the compatible family of degree-4 Scholl representations $\rho_{\ell}$ associated with a space $S$ of weight $\kappa> 2$ noncongruence cusp forms satisfying Quaternion Multiplications over a biquadratic field $K$. It is shown that when either $K$ is totally real or $\kappa$ is odd, $\rho_\ell$ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for $GL_4(\mathbb Q)$. Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in $S$ and those of certain automorphic forms via the three-term Atkin and Swinnerton-Dyer congruences.

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